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{\bf Anders Sune Pedersen}
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{\bf Complete and Almost Complete Minors in Double-Critical $8$-Chromatic Graphs}
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A connected $k$-chromatic graph $G$ is said to be {\it
  double-critical} if for all edges $uv$ of $G$ the graph $G - u - v$
is $(k-2)$-colourable. A longstanding conjecture of Erd\H{o}s and
Lov\'asz states that the complete graphs are the only double-critical
graphs. Kawarabayashi, Pedersen and Toft [\emph{Electron.\ J.\
  Combin.}, 17(1): Research Paper 87, 2010] proved that every
double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$
minor. It remains unknown whether an arbitrary double-critical
$8$-chromatic graph contains a $K_8$ minor, but in this paper we prove
that any double-critical $8$-chromatic contains a minor isomorphic to
$K_8$ with at most one edge missing. In addition, we observe that any
double-critical $8$-chromatic graph with minimum degree different from
$10$ and $11$ contains a $K_8$ minor.

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